Infinity - Notes on Behaviour at Infinity The behaviour of...

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Unformatted text preview: Notes on Behaviour at Infinity The behaviour of a function y (t) is determined by the limit of y (t) as t → ∞. Elementary Functions The basic functions we need are powers of t and exponentials of t. These have the following limits. lim tα = t→∞ ∞ if α > 0, 0 if α < 0 lim eαt = t→∞ ∞ if α > 0, 0 if α < 0 Of course if these functions are multiplied by a function, the sign of the limit can change: 3t2 → ∞ but −5t7 → −∞. Sums of Functions Suppose we have y (t) = y1 (t) + y2 (t). Then in general we have lim y (t) = lim y1 (t) + lim y2 (t), t→∞ t→∞ t→∞ and even when the limits are infinite we can use the rules that for any real number a, ∞ + a = ∞ and −∞ + a = −∞, together with the rules that ∞ + ∞ = ∞ and −∞ + (−∞) = −∞. A problem occurs when we get ∞ + (−∞) which is not defined. In this case we need to compare the functions and determine which function grows to infinity faster and is therefore the dominant term. If y1 (t) dominates y2 (t) for large t, we express this by writing y1 (t) y2 (t) as t → ∞. And in this case lim y (t) = lim y1 (t), t→∞ t→∞ so the behaviour of y (t) is determined by the behaviour of the dominant term (and we can ignore the other terms). For exponentials and polynomials we have the follows dominance relations. • For α, β > 0, eαt tβ as t → ∞. • For α > β > 0, eαt eβt as t → ∞. • For α > β > 0, tα tβ as t → ∞. Examples lim −et + 11t−4 = −∞ t→∞ lim t2 + t−2 = +∞ t→∞ lim 1 + t−1 + t−2 + e−t = 1 t→∞ lim 3et − t2 − t7 = lim 3et = +∞ t→∞ t→∞ lim 5et − e2t = lim −e2t = −∞ t→∞ t→∞ lim t3 − t2 − t + 7 − e−8t t→∞ +∞ 4t 2t lim Ce − 3e = −∞ t→∞ −∞ = lim t3 = +∞ t→∞ if C > 0, if C = 0, if C < 0 +∞ if y0 > −2, t −2 lim (y0 + 2)e + t + 3 = 3 if y0 = −2, t→∞ −∞ if y0 < −2 lim (y0 + 2)e−2t + 4 = 4 t→∞ 1 ...
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